Three-valued formalization of logic programming. Is it needed?

Abstract

The central issue of this paper concerns the truth value undefined in Przymusinsi's 3-valued formalization of nonmonotonic reasoning and logic programming. We argue that this formalization can lead to the problem of unintended semantics and loss of disjunctive information. We modify the formalization by proposing two general principles for logic program semantics: justifiability and minimal undefinedness. The former is shown to be a general property for almost all logic program semantics, and the latter requires the use of the undefined only when it is necessary. We show that there are three types of information embedded in the undefined: the disjunctive, the factoring, and the 'difficult-to-be-assigned'. In the modified formalization, the first two can be successfully identified and branched into multiple models. This leaves only the 'difficult-to-be-assigned' as the undefined. It is shown that the truth value undefined is needed only for a very special type of programs whose practicality is yet to be evidenced.